Answer :
To solve the problem of factoring the polynomial [tex]\( V(x) = \frac{L}{6} x^4 - m^2 x^2 \)[/tex] and determining the locations of the [tex]\( x \)[/tex]-intercepts, we will follow these steps:
### Step 1: Factoring [tex]\( V(x) \)[/tex]
Given the polynomial:
[tex]\[ V(x) = \frac{L}{6} x^4 - m^2 x^2 \][/tex]
We first note that we can factor out a common term, [tex]\( x^2 \)[/tex], from both terms in the polynomial:
[tex]\[ V(x) = x^2 \left( \frac{L}{6} x^2 - m^2 \right) \][/tex]
Now we have:
[tex]\[ V(x) = x^2 \left( \frac{L}{6} x^2 - m^2 \right) \][/tex]
To simplify further, we leave it in this factored form since it is compact and already shows the expression in terms of products.
### Step 2: Finding the [tex]\( x \)[/tex]-intercepts
The [tex]\( x \)[/tex]-intercepts occur where [tex]\( V(x) \)[/tex] equals zero:
[tex]\[ V(x) = 0 \][/tex]
Thus,
[tex]\[ x^2 \left( \frac{L}{6} x^2 - m^2 \right) = 0 \][/tex]
We solve this equation by setting each factor to zero separately:
1. [tex]\( x^2 = 0 \)[/tex]
2. [tex]\( \frac{L}{6} x^2 - m^2 = 0 \)[/tex]
#### Solving [tex]\( x^2 = 0 \)[/tex]
This implies:
[tex]\[ x = 0 \][/tex]
#### Solving [tex]\( \frac{L}{6} x^2 - m^2 = 0 \)[/tex]
To solve this, we isolate [tex]\( x^2 \)[/tex]:
[tex]\[ \frac{L}{6} x^2 = m^2 \][/tex]
Multiplying both sides by [tex]\( \frac{6}{L} \)[/tex] to solve for [tex]\( x^2 \)[/tex]:
[tex]\[ x^2 = \frac{6m^2}{L} \][/tex]
Taking the square root of both sides, we find:
[tex]\[ x = \pm \sqrt{\frac{6m^2}{L}} \][/tex]
This simplifies to:
[tex]\[ x = \pm \sqrt{6} m \sqrt{\frac{1}{L}} \][/tex]
### Summary of [tex]\( x \)[/tex]-intercepts
Combining these results, the [tex]\( x \)[/tex]-intercepts are:
[tex]\[ x = 0, x = -\sqrt{6} m \sqrt{\frac{1}{L}}, x = \sqrt{6} m \sqrt{\frac{1}{L}} \][/tex]
### Final Answer
Thus, the factored form of [tex]\( V(x) \)[/tex] and the locations of the [tex]\( x \)[/tex]-intercepts are:
Factored Form:
[tex]\[ V(x) = x^2 \left( \frac{L}{6} x^2 - m^2 \right) \][/tex]
[tex]\( x \)[/tex]-intercepts:
[tex]\[ x = 0, \quad x = -\sqrt{6} m \sqrt{\frac{1}{L}}, \quad x = \sqrt{6} m \sqrt{\frac{1}{L}} \][/tex]
### Step 1: Factoring [tex]\( V(x) \)[/tex]
Given the polynomial:
[tex]\[ V(x) = \frac{L}{6} x^4 - m^2 x^2 \][/tex]
We first note that we can factor out a common term, [tex]\( x^2 \)[/tex], from both terms in the polynomial:
[tex]\[ V(x) = x^2 \left( \frac{L}{6} x^2 - m^2 \right) \][/tex]
Now we have:
[tex]\[ V(x) = x^2 \left( \frac{L}{6} x^2 - m^2 \right) \][/tex]
To simplify further, we leave it in this factored form since it is compact and already shows the expression in terms of products.
### Step 2: Finding the [tex]\( x \)[/tex]-intercepts
The [tex]\( x \)[/tex]-intercepts occur where [tex]\( V(x) \)[/tex] equals zero:
[tex]\[ V(x) = 0 \][/tex]
Thus,
[tex]\[ x^2 \left( \frac{L}{6} x^2 - m^2 \right) = 0 \][/tex]
We solve this equation by setting each factor to zero separately:
1. [tex]\( x^2 = 0 \)[/tex]
2. [tex]\( \frac{L}{6} x^2 - m^2 = 0 \)[/tex]
#### Solving [tex]\( x^2 = 0 \)[/tex]
This implies:
[tex]\[ x = 0 \][/tex]
#### Solving [tex]\( \frac{L}{6} x^2 - m^2 = 0 \)[/tex]
To solve this, we isolate [tex]\( x^2 \)[/tex]:
[tex]\[ \frac{L}{6} x^2 = m^2 \][/tex]
Multiplying both sides by [tex]\( \frac{6}{L} \)[/tex] to solve for [tex]\( x^2 \)[/tex]:
[tex]\[ x^2 = \frac{6m^2}{L} \][/tex]
Taking the square root of both sides, we find:
[tex]\[ x = \pm \sqrt{\frac{6m^2}{L}} \][/tex]
This simplifies to:
[tex]\[ x = \pm \sqrt{6} m \sqrt{\frac{1}{L}} \][/tex]
### Summary of [tex]\( x \)[/tex]-intercepts
Combining these results, the [tex]\( x \)[/tex]-intercepts are:
[tex]\[ x = 0, x = -\sqrt{6} m \sqrt{\frac{1}{L}}, x = \sqrt{6} m \sqrt{\frac{1}{L}} \][/tex]
### Final Answer
Thus, the factored form of [tex]\( V(x) \)[/tex] and the locations of the [tex]\( x \)[/tex]-intercepts are:
Factored Form:
[tex]\[ V(x) = x^2 \left( \frac{L}{6} x^2 - m^2 \right) \][/tex]
[tex]\( x \)[/tex]-intercepts:
[tex]\[ x = 0, \quad x = -\sqrt{6} m \sqrt{\frac{1}{L}}, \quad x = \sqrt{6} m \sqrt{\frac{1}{L}} \][/tex]
